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In [[statistics]], an '''approximate entropy''' ('''ApEn''') is a technique used to quantify the amount of regularity and the [[unpredictability]] of fluctuations over [[time-series]] data.<ref name="Pincus1991">
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{{cite journal
|last1=Pincus |first1=S. M.
|last2=Gladstone |first2=I. M.
|last3=Ehrenkranz
|first3=R. A.
|title=A REGULARITY STATISTIC FOR MEDICAL DATA ANALYSIS
|journal=[[Journal of Clinical Monitoring and Computing]]
|year=1991 |volume=7 |issue=4 |pages=335–345
|DOI=10.1007/BF01619355
}}
</ref>
 
For example, there are two series of data:
: series 1: (10,20,10,20,10,20,10,20,10,20,10,20...), which alternates 10 and 20.
 
: series 2: (10,10,20,10,20,20,20,10,10,20,10,20,20...), which has either a value of 10 or 20, chosen randomly, each with probability 1/2.
 
[[Moment (mathematics)|Moment statistics]], such as [[mean]] and [[variance]], will not distinguish between these two series. Nor will [[rank order]] statistics distinguish between these series. Yet series 1 is "perfectly regular"; knowing one term has the value of 20 enables one to predict with certainty that the next term will have the value of 10. Series 2 is randomly valued; knowing one term has the value of 20 gives no insight into what value the next term will have.
 
Regularity was originally measured by exact regularity statistics, which has mainly centered around various entropy measures.<ref name="Pincus1991" />
However, accurate entropy calculation requires vast amounts of data, and the results will be greatly influenced by system noise,<ref name="Pincus21991">
{{cite journal
|last1=Pincus |first1=S. M.
|title=Approximate entropy as a measure of system complexity
|journal=[[Proceedings of the National Academy of Sciences]]
|year=1991 |volume=88 |issue=6 |pages=2297–2301
|PMID=11607165 |doi=10.1073/pnas.88.6.2297 |pmc=51218
}}
</ref> therefore it is not practical to apply these methods to experimental data. ApEn was developed by [[Steve M. Pincus]] to handle these limitations by modifying an exact regularity statistic, [[Kolmogorov-Sinai entropy]].  ApEn was initially developed to analyze medical data, such as heart rate,<ref name="Pincus1991" /> and later spread its applications in [[finance]],<ref name="Pincus2004">
{{cite journal
|last1=Pincus |first1=S.M.
|last2=Kalman |first2=E.K.
|title=Irregularity, volatility, risk,  and financial market time series
|journal=[[Proceedings of the National Academy of Sciences]]
|year=2004 |volume=101 |issue=38 |pages=13709–13714 |PMID=15358860 |doi=10.1073/pnas.0405168101 |pmc=518821
}}
</ref> [[psychology]],<ref name="Pincus1994">
{{cite journal
|last1=Pincus |first1=S.M.
|last2=Goldberger |first2=A.L.
|title=Physiological time-series analysis: what does regularity quantify?
|journal=[[The American journal of physiology]]
|year=1994 |volume=266 |issue=4 |pages=1643–1656 |PMID=8184944
}}
</ref> and [[human factors engineering]].<ref name="humanfactor">
{{cite journal
|last1=McKinley |first1=R.A.
|last2=McIntire |first2=L.K.
|last3=Schmidt |first3=R
|last4=Repperger |first4=D.W.
|last5=Caldwell |first5=J.A.
|title=Evaluation of Eye Metrics as a Detector of Fatigue
|journal=[[HUMAN FACTORS]]
|year=2011 |volume=53 |issue=4 |pages=403–414
|DOI=10.1177/0018720811411297
}}
</ref>
 
==The algorithm==
An implementation on Physionet <ref name=physionet>[http://physionet.org/physiotools/ApEn/]</ref>, which is based on Pincus <ref name="Pincus21991"/> use use <math>< r </math> whereas the original article uses <math> \le r </math>. This should only matter (i.e. make a difference to the result) in artificially constructed examples:
<math>\text{Step 1}</math>: Form a time series of data <math>\ u(1), u(2),\ldots, u(N)</math>. These are <math>\text{N}</math> raw data values from measurement equally spaced in time.
 
<math>\text{Step 2}</math>: Fix <math>\ m </math>, an [[integer]], and <math>\ r</math>, a [[positive number|positive]] [[real number]]. The value of <math> \ m </math> represents the length of compared run of data, and <math> \ r </math> specifies a filtering level.
 
<math>\text{Step 3}</math>: Form a sequence of vectors <math>\mathbf{x}(1)</math>,<math>\mathbf{x}(2),\ldots,\mathbf{x}(N-m+1)</math>, in <math>\mathbf{R}^{m}</math>, real <math>\ m</math>-dimensional space defined by <math>\mathbf{x}(i) = [u(i),u(i+1),\ldots,u(i+m-1)]</math>.  
 
<math>\text{Step 4}</math>: Use the sequence <math>\mathbf{x}(1)</math>,<math>\mathbf{x}(2),\ldots,\mathbf{x}(N-m+1)</math> to construct, for each <math> \ i </math>, <math> 1 \le i \le N-m+1 </math>
 
:<math> C_i^m (r)=(\text{number of } x(j) \text { such that } d[x(i),x(j)] < r)/(N-m+1) \, </math>
 
in which <math>\ d[x, x^*]</math> is defined as
 
:<math> d[x,x^* ]=\max_a |u(a)-u^*(a)| \,</math>
 
The <math> \ u(a) </math> are the  <math> \text {m} </math> [[scalar (mathematics)|scalar]] components of <math> \mathbf{x} </math>. <math> \ d </math> represents the distance between the [[vector (mathematics and physics)|vectors]] <math> \mathbf{x}(i) </math> and <math>\mathbf{x}(j) </math>, given by the maximum difference in their respective scalar components. Note that <math>j</math> takes on all values, so the match provided when <math>i=j</math> will be counted (the subsequence is matched against itself).
 
<math>\text{Step 5}</math>: Define
:<math> \Phi ^m (r) = (N-m+1)^{-1} \sum_{i=1}^{N-m+1}log(C_i^m (r))</math>,
 
<math>\text{Step 6}</math>: Define approximate entropy <math>\ (\mathrm{ApEn})</math> as
 
:<math>\ \mathrm{ApEn} = \Phi ^m (r) - \Phi^{m+1} (r). </math>
 
where <math>\ log </math> is the natural logarithm, for <math>\ m </math> and <math>\ r </math> fixed as in Step 2.  
 
Parameter selection: typically choose <math>\ m=2 </math> or <math>\ m=3 </math>, and <math>\ r </math> depends greatly on the application.
 
==The interpretation==
The presence of repetitive patterns of fluctuation in a time series renders it more predictable than a time series in which such patterns are absent. ApEn reflects the likelihood that ''similar'' patterns of observations will not be followed by additional ''similar'' observations.<ref>
{{cite journal
|last1=Ho |first1=K. K.
|last2=Moody |first2=G. B.
|last3=Peng 
|first3=C.K.
|last4=Mietus |first4=J. E.
|last5=Larson |first5=M. G.
|last6=levy |first6=D
|last7=Goldberger |first7=A. L.
|title=Predicting survival in heart failure case and control subjects by use of fully automated methods for deriving nonlinear and conventional indices of heart rate dynamics
|journal=[[Circulation (journal)|Circulation]]
|year=1997 |volume=96 |issue=3 |pages=842–848
|PMID=9264491
}}
</ref> A time series containing many repetitive patterns has a relatively small ApEn; a less predictable process has a higher ApEn.
 
==One example==
[[File:Heartrate.jpg|thumb|Illustration of the Heart Rate Sequence]]
Suppose <math>\ N=51 </math>, and the sequence consists of 51 samples of heart rate equally spaced in time:
 
:<math> \ S_N = \{85, 80, 89, 85, 80, 89, \ldots\} </math>
 
(i.e., the sequence is periodic with a period of 3). Let's choose <math>\ m=2 </math> and <math>\ r=3</math> (the values of <math>\ m </math> and <math>\ r </math> can be varied without affecting the result).  
 
Form a sequence of vectors:
:<math>\mathbf{ x}(1) = [u(1) \,u(2)]=[85\, 80]</math>
:<math>\mathbf{ x}(2) = [u(2)\, u(3)]=[80\, 89]</math>
:<math>\mathbf{ x}(3) = [u(3)\, u(4)]=[89\, 85]</math>
:<math>\mathbf{ x}(4) = [u(4)\, u(5)]=[85\, 80]</math>…
 
Distance is calculated as follows:
 
:<math>\ d[\mathbf{x}(1), \mathbf{x}(1)]=\max_a |u(a)-u^*(a)|=0<r=3 </math>
 
Note <math>\ |u(2)-u(3) |>|u(1)-u(2) |</math>, so
 
:<math>\ d[\mathbf{x}(1), \mathbf{x}(2)]=\max_a |u(a)-u^*(a)|=|u(2)-u(3)|=9>r=3 </math>
Similarly,
:<math>\ d[\mathbf{x}(1), \mathbf{x}(3)]=|u(2)-u(4) |=5>r </math>
:<math>\ d[\mathbf{x}(1), \mathbf{x}(4)]=|u(1)-u(4) |=|u(2)-u(5) |=0<r </math>
Therefore, <math>\mathbf{ x}(j)\text{s}</math>  such that <math>\ d[\mathbf{x}(1),\mathbf{x}(j)]\le r </math> include <math> \mathbf{x}(1), \mathbf{x}(4), \mathbf{x}(7),\ldots,\mathbf{x}(49)</math>, and the total number is 17.  
 
:<math>\ C_1^2 (3)=\frac{17}{50}</math>
:<math>\ C_2^2 (3)=\frac{17}{50}</math>
:<math>\ C_3^2 (3)=\frac{16}{50}</math>
:<math>\ C_4^2 (3)=\frac{17}{50}\  \ldots</math>
 
Please note in Step 4, for <math> \mathbf{x}(i) </math>,  <math>\ 1 \le i \le N-m+1 </math>. So the <math>\mathbf{x}(j)\text{s}</math> such that <math>\ d[\mathbf{x}(3),\mathbf{x}(j)] < r </math> include <math> \mathbf{x}(3), \mathbf{x}(6), \mathbf{x}(9),\ldots,\mathbf{x}(48)</math>, and the total number is 16.
 
:<math>\Phi^2 (3)=(50)^{-1} \sum_{i=1}^{50}C_i^2(3)\approx0.3336 </math>
 
Then we repeat the above steps for m=3. First form a sequence of vectors:
:<math>\mathbf{ x}(1) = [u(1)\, u(2)\, u(3)]=[85\, 80\, 89]</math>
:<math>\mathbf{ x}(2) = [u(2)\, u(3)\, u(4)]=[80\, 89\, 85]</math>
:<math>\mathbf{ x}(3) = [u(3)\, u(4)\, u(5)]=[89\, 85\, 80]</math>
:<math>\mathbf{ x}(4) = [u(4)\, u(5)\, u(6)]=[85\, 80\, 89]</math>…
 
By calculating distances between vector <math>\mathbf{x}(i), \mathbf{x}(j), 1 \le i \le 49 </math>, we find the vectors satisfying the filtering level have the following characteristic:
:<math>\ d[\mathbf{x}(i)\,\mathbf{x}(i+3)]=0<r </math>
Therefore, 
:<math>\ C_1^3 (3)=\frac{17}{49}</math>
:<math>\ C_2^3 (3)=\frac{16}{49}</math>
:<math>\ C_3^3 (3)=\frac{16}{49}</math>
:<math>\ C_4^3 (3)=\frac{17}{49}\  \ldots</math>
 
:<math>\Phi^3 (3)=(49)^{-1} \sum_{i=1}^{49} C_i^3(3)\approx0.3336 </math>
 
Finally,
:<math> \mathrm{ ApEn}=\log(\Phi^2 (3))-\log(\Phi^3 (3))\approx0.000033</math>
 
The value is very small, so it implies the sequence is regular and predictable, which is consistent with the observation.
 
==Advantages==
The advantages of ApEn include:<ref name="Pincus21991" />
*Lower computational demand. ApEn can be designed to work for small data samples (n < 50 points) and can be applied in real time.
*Less effect from noise. If data are noisy, the ApEn measure can be compared to the noise level in the data to determine what quality of true information may be present in the data.
 
==Applications==
ApEn has been applied to classify EEG in psychiatric diseases, such as schizophrenia,<ref name="Sabeti2009">{{cite journal
|last1=Sabeti |first1=Malihe
|title=Entropy and complexity measures for EEG signal classification of schizophrenic and control participants
|journal=[[Artificial Intelligence in Medicine]]
|year=2009 |volume=47 |issue=3 |pages=263–274
|PMID=19403281 |doi=10.1016/j.artmed.2009.03.003
}}
</ref> epilepsy,<ref name="Yuan2011">{{cite journal
|last1=Yuan |first1=Qi
|title=Epileptic EEG classification based on extreme learning machine and nonlinear features
|journal=[[Epilepsy Research]]
|year=2011 |volume=96 |issue=1-2 |pages=29–38
|PMID=21616643 |doi=10.1016/j.eplepsyres.2011.04.013
}}
</ref> and addiction.<ref name="Yun2012">{{cite journal
|last1=Yun |first1=Kyongsik
|title=Decreased cortical complexity in methamphetamine abusers
|journal=[[Psychiatry Research: Neuroimaging]]
|year=2012
|PMID=22445216
|doi=10.1016/j.pscychresns.2011.07.009
|volume=201
|issue=3
|pages=226–32
}}
</ref>
 
==Limitations==
The ApEn algorithm counts each sequence as matching itself to avoid the occurrence of ln(0) in the calculations. This step might cause bias of ApEn and this bias causes ApEn to have two poor properties in practice:<ref>
{{cite journal
|last1=Richman |first1=J.S.
|last2=Moorman |first2=J.R.
|title=Physiological time-series analysis using approximate entropy and sample entropy
|journal=[[American journal of physiology. Heart and circulatory physiology]]
|year=2000 |volume=278 |issue=6 |pages=2039–2049 |PMID=10843903
}}
</ref>
*First, ApEn is heavily dependent on the record length and is uniformly lower than expected for short records.  
*Second, it lacks relative consistency. That is, if ApEn of one data set is higher than that of another, it should, but does not, remain higher for all conditions tested.
 
==See also==
*[[Recurrence quantification analysis]]
 
==References==
{{reflist}}
 
[[Category:Time series analysis]]
[[Category:Entropy and information]]

Latest revision as of 22:03, 29 October 2014

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To improve muscular mass in the most beneficial way possible, you should warm-up prior to every single workout. Once your muscle groups become stronger, they'll practical experience a great deal of additional tension causing them to be susceptible to trauma. By warming up, these personal injuries could be averted. Before you decide to lift anything heavy, do 5-ten minutes of light cardio, combined with several light-best weight gainer protein supplement or intermediate sets of your normal exercises.

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Carbohydrate food are important to muscle development. Carbohydrate food are what your system needs to offer the power for routines. Should you don't consume an adequate amount of them the body can break down proteins for electricity alternatively. Try and get sufficient carbohydrates for you can get the most from your workout routines.

Change your workout program from 7 days to week. If you do a similar workout routine time in and day trip, you will find a greater probability that you receive fed up, and in all likelihood end your exercises completely. Ensure that you do diverse exercises and work out various muscle tissues every time you workout. If you alter your programs once in awhile, you can expect to remain interested and determined longer.

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The key objective associated with a muscle development workout is to boost your power. You need to see a steady surge in the level of weight you may lift up after a while. In fact, if you are just starting out, you need to be capable of raising about 5 percent more each two workout routines. You have to reassess your software when your progress is slow than this. Should you feeling you might have not received power since your previous work out, you possibly will not have made it possible for oneself enough recovery time.

Muscle mass building isn't generally an attempt in becoming exceedingly heavy. Various muscle mass building regimens will sculpt the body differently. In the event you want very large muscle groups than you can expect to at some point probably might need some supplements in addition to your daily diet and workouts.

Make sure that you are eating the amount of calories that your system needs. There are a variety of on-line calculators to assist you estimate what your calorie require is, for the way very much muscle mass you want to create. By utilizing this type of calculator, you can quickly know what alterations should be designed to what you eat for preferred leads to be received.

You may inform your muscle mass building program works well should you be getting more robust from few days to 7 days. The more powerful you get, the heavier the body weight it will be possible to control on your education. In the event you examine while you are just start, you need to see on your own lifting five percent a lot more than you may have two workout sessions in the past. This craze ought to be regular. If you're not meeting targets, rethink your program. If you think as though you possess significantly less strength than during your final exercise, you may not have made it possible for the muscles to totally recuperate.

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